LIBRARY OF CONGRESS. 



Shelf .T_ilT 

UNITED STATES OF AMERICA. 



/ / / 



INTERMEDIATE COURSE 



IN 



MECHANICAL DRAWING, 



WILLIAM H. THORNE, 

Director of the Drawing School of the Franklin Institute of Philadelphia. 

Oi ^ f<r COPYRIGHT 

,141889 

/ 

PUBLISHED BY 

WILLIAMS & BROWN, 
N. E. Cor. Chestnut and Tenth Sts., Philadelphia. 



^^iii 



^^^ 



Entered according to Act of Congress, in the year 1889, by 

WILLIAM H. THORNE, 
In the Office of the Librarian of Congress, at Washington. 






My intention, in arranging this Intermediate Course in Mechanical 
Drawing, has been to present the subject of Orthographic Projection in a con- 
cise, logical and comprehensive manner, keeping theory always in view, but, 
at the same time, doing everything according to methods which practical 
experience has proven to be the easiest, most accurate, useful and readily 
interpreted. 

The endeavor has been to avoid giving, on the one hand, a mass of 
definitions, rules, theorems and analyses, which are so readily forgotten, and, 
on the other hand, minute directions for every line, which would make the 
study a mere copying process ; but to encourage and induce the interest, close 
attention and thought of the student, and thus bring about a thorough compre- 
hension of the theory and principles and a correct training for the practice of 
Mechanical Drawing, so that all the apparently new conditions, which are con- 
stantly arising, can be analyzed, not by reference to text-book rules of doubtful 
applicability, but by the exercise of the individual reasoning powers. 

WM. H. THORITE. 

GowEN Ave., Mount Airy, 

Philadelphia, 1889. 



INTRODUCTION. 

As has been explained in the Junior Course, the purpose of Mechanical 
Drawing is to give such an illustration of a required object as to enable it to be 
accurately and definitely built from the drawing alone. To accomplish this 
purpose, the drawing must contain a sufficient number of views of the object 
to show the true size of every feature necessary for the information of the 
person who is to build it. It is not sufficient that the lines and surfaces be 
fully determined geometrically, but their dimensions and relative positions must 
also be shown in the way to be most easily understood and to require the least 
eftbrt of thought in the interpretation of the drawing. The thinking must be done 
by the draughtsman, at least as to the form, size and purpose of the structure, 
and the drawing should embody that thought in every particular. Mechanical 
Drawing is an embodiment of thought, and, largely, of original thought, com- 
bined with technical knowledge and manual training. Hence, anything like 
copying should be studiously avoided when the theory and principles are being 
studied, and the mind should be trained from the start to work from a concep- 
tion, an idea, and to put that idea on paper so as to be understood. The 
imagination, or what might be called the scientific imagination, must be con- 
tinually on the alert, because the hand is constantly called upon to do something 



6 INTERMEDIATE COURSE. 

that it has never done before, the diiferent conditions and combinations which 
are constantly arising being infinite in their variety. Drawing from models 
should also be avoided. They may be used at the beginning for the purpose 
of giving a mental conception of the object before the student's mind has been 
trained to form a clear idea of it from a verbal description, or before his inven- 
tive faculties have been sufficiently developed to enable him to originate one in 
his own mind; but he should soon be made to realize the proper sequence of 
the art, which is : first, the conception, either original or derived from another; 
and, second, the representation of that conception by means of a drawing in 
such a clear, explicit and accurate manner as to enable an artisan to produce it 
in the concrete. The habit of drawing from models is the reverse of this, and 
does not tend to properly develop the imagination or train the mind for original 
work or produce good methods. Copying from other drawings must only be 
done for the purpose of gaining experience of the principles involved and the 
style of execution ; but in every case another drawing should be made, with the 
proportions and conditions so changed as to compel an exercise of original 
thought. All drawings should be inked, and inked properly, imaginary lines 
being distinguished from actual lines. After spending time, work and thought 
on a drawing, it is simply folly to leave it in a condition which is unintelligible 
to any one but the author, and which will soon become so to him. The addi- 
tional time required to ink, finish and dimension the drawing is amply repaid 



MECHANICAL DRAWING. < 

by the clearness, permanence and usefulness of the result. N'ever make a draw- 
ing, not even a purely theoretical one, without using definite measurements, and 
always put on the essential dimensions. This habit should be cultivated from 
the start, as it tends to accuracy of thought and workmanship, and is a training 
for the proper selection of the parts requiring dimensions and of the best places 
to mark them. 

The dimension lines and all imaginary lines, or those which do not repre- 
sent parts of the object itself, but serve as bases or lines of reference, or which 
are necessarily used in the construction of the drawing and" the preservation of 
which would be desirable, should be distinguished from the black lines of the 
object in such manner as not to interfere with the clearness of the latter. In 
other words, the object should stand out boldly and clearly to permit a ready 
general comprehension of it, while the detailed information concerning it should 
be kept somewhat in the background. The importance of this is not so appar- 
ent in the drawing of the simple geometrical solids in the following problems, 
but it is very great in all work of any intricacy. 

All the devices which are used by intelligent and experienced draughtsmen 
to add to the usefulness, clearness and beauty of their work, should be employed 
by the student, in order that their use may become, in a manner, intuitive, and 
thus leave the mind free for concenti'ation on the construction. 



INTERMEDIATE COURSE. 



PROJECTIONS, 

The Junior Course has shown that Mechanical Drawings are made upon 
the theory that the imaginary object is surrounded by imaginary planes perpen- 
dicular to each other, and that lines perpendicular to these planes are projected 
from each point of the object to each of the planes, and that the points where 
the projecting lines pierce the planes are the projections on these planes of the 
points of the object, and that these planes are then supposed to be revolved 
upon their intersections to bring them all into one plane, — that of the paper. 
The projections on each of these planes thus form a distinct view from one 
direction, and give the appearance which the object would present if viewed 
from an infinite distance in a direction perpendicular to the plane, the plane 
being between the object and the point of sight. The revolution of these planes 
into the plane of the paper brings the several views into the most convenient 
and sensible positions in relation to each other. It brings the Plan or Top View 
above, the Right Side View to the right, the Left Side View to the left, and any 
oblique view immediately adjacent to the part it represents. These are the 
positions which experience in the making of intricate drawings has proven to 
be the clearest and most manageable, and to be preferable to the opposite 
system, which imagines the object to be between the plane of projection and 



MECHANICAL DRAWING, U 

the point of sight, with the result of locating the views in the opposites of these 
positions ; that is, a view of the right side would be at the left, a view of the 
top would be underneath, and so on. Advocates of the latter system rarely 
adhere rigidly to it, and, in drawing oblique views, almost invariably violate it, 
and this fact is one of the strongest reasons against using it. Apart from its 
inconvenience, there is no vital objection to it provided that it is rigidly adhered 
to, because then the location of a view will immediately and positively indicate 
which side of the object it represents; but if the two systems are both used in 
the same drawing, doubt and possibly errors will result. To those acquainted 
with Descriptive Geometry, the method here advocated and invariably employed 
is the use of the third angle and not the tirst. 

The actual indication of the planes of projection, by showing their inter- 
sections or the axes about which they are supposed to have been revolved into 
the plane of the paper, has already been abandoned in the Junior Course, and 
will not be used at all in this Intermediate Course, as it is desirable that the mind 
should be trained to recognize the relation of one view to another without the 
intervention of these axes of projection. 

The number of possible views of the same object is infinite, and only such 
as will show in the simplest manner the essential dimensions and form should 
be selected. It frequently occurs, however, that one detail or unit of a structure 
is oblique to the main body, necessitating oblique views. Figs. 59 and 60 are 



10 INTERMEDIATE COURSE. 

given as examples of such cases, and also for the purpose of showing the variety 
of different views which can be made and their proper relation to each other. 



TECHNICALITIES, 

Each Plate represents a sheet of drawing paper 16 by 21 inches, with 
margin lines 15 by 20. The Figures are one-fourth size, but are to be drawn 
full size. 

All the lines representing the Object are to be inked black, those repre- 
senting visible parts being full lines, and those representing hidden parts being 
composed of short dots, as shown by the full lines and short-dotted lines in the 
Plates. Shade Lines are to be used in all cases, and to be located on the 
theory that the light falls upon the drawing from the upper left hand at an 
angle of 45°, and produces the same effect on all the different views, namely: of 
making the lower and right-hand edges shaded. 

Circles and curves are to be inked first, and each shaded immediately, then 
all the straight Jine and dotted lines, and finally the heavi/ shade lines. 

After completing the black lines, ink all the Centre Lines or axes of 
symmetry and any important lines of reference, blue; lastly, ink all the dimen- 
sion lines and any construction lines used in obtaining the lines of the object, 



MECHANICAL DRAWING. 11 

the preservation of which is desirable, red. The dimensions and the arrow- 
heads or points at the extremities of the dimension lines should be black. 

In the accompanying Plates, the Figures being printed entirely in black, 
the blue centre lines are represented by long dots, the red construction lines by 
long-and-short dots, and the red dimension lines can not be mistaken. In 
making a drawing, however, the blue centre lines and red construction lines 
must not be dotted, but made full lines. 

For directions as to handling instruments and materials, refer to pages 5, 6, 
8, 30, 31 and 32 of the Junior Course. 

Adopt some neat style of lettering that can be easily and quickly executed, 
and form the habit of alwaj^s putting a signature and date as well as a title upon 
all drawings intended for practical use, in order to ensure their identification; 
but never make these conspicuous, because the primary object is to show the 
structure, to which everything should be made subservient. 

Neat execution and artistic effect are desirable qualities in a drawing, but 
correctness, clearness, and an emphatic, unquestionable expression of precisely 
what is meant, is still more important. It is the beauty of the conception and 
the exactness with which the result can be produced in the concrete, which con- 
stitutes the beauty of a mechanical drawing, and, therefore, any attempt at 
scenic effect is not only in bad tas'te, but often detracts from its usefulness. If, 
however, a picture of the structure is required for the information of those who 



12 INTERMEDIATE COURSE. 

are unable to understand a working drawing, or to conceive of what it repre- 
sents, then artistic effect becomes the principal object. 



OBLIQUE VIEWS. 



PLATE 9. 

Fig. 59. To make a mechanical drawing of a structure -which is oblique to the 
vertical -planes of projection. 

This condition would, of course, occur only when the oblique structure was 
bxit a part of a complicated whole, the more important features of which were 
parallel to the planes of projection. 

Let the oblique structure have a horizontal rectangular base %h" long and 
If" wide, the longitudinal centre line of the base making an angle of 30° with 
the front vertical plane. Let the two sides and two ends of the structure incline 
at 60° with the base, and let the base, sides and ends be \" thick. 

In the first place, determine a convenient position on the drawing paper for 
the Plan or Top View, and locate the centre p of the base, through which draw 
the vertical line ab for the trace of a vertical plane perpendicular to the front 



MECHANICAL DRAWING. 13 

elevation, and the horizontal line cd for the trace of a vertical plane parallel to the 
front elevation. These two lines then represent two vertical planes perpendicular 
to each other and to the planes of projection, and passing through the centre 
of the structure, and will serve as bases from which to locate the points. 

Through the central point p, at the given angle, 30° with cd, draw gh for 
the longitudinal centi-e line of the base, and perpendicular to gh draw ef for the 
transverse centime line. On these centre lines, about the central point p, draw 
the base 3|" by If", as given. We now have the traces of four vertical planes 
intersecting at the centre or axis of the structure, two of which are parallel to 
the planes of projection and two normal to the sides of the structure, and we 
have the outline of the base on a horizontal plane. Before we can obtain the 
Front and Side Elevations, which will be oblique views, we must first draw the 
projections upon planes parallel to the structui'e, because only upon such planes 
does it appear in its true form and size. Hence, project an End Elevation, B, 
of the base by drawing a line equal to its width perpendicular to gh, and from 
the extremities of this line draw the inclined sides, and, parallel with these sides 
and the base and \" from them, draw the interior lines for their thickness. 

It is a fundamental principle of mechanical drawing that each view should 
be made to give all the useful information than it can in the clearest manner. 
In the End View which we have just drawn, the additional iTiformation beyond 
that given by the Plan is the inclination of the sides and the thickness of the 



14 INTERMEDIATE COURSE. 

sides and base. It is evident that this thickness can be shown more clearly by 
representing the structure as cut through to expose it, than by indicating it by 
dotted lines. The representation of such a cut is called a Section, and is made 
by equidistant fine black lines at 45° with the principal side. The distance 
between the lines varies from -^" to \" , according to the scale of the drawing 
and the size of the object, and is determined by judgment and taste. The 
Section is supposed to be on the central plane unless otherwise indicated, and is 
always on a plane parallel to the plane of projection of the view showing it. In the 
present instance, it is a Vertical Section on plane ef. 

From this End View, B, and the Plan of the base, A, project an auxiliary 
Front View, C, by drawing the trace, e'f, of the vertical plane, ef, and projecting 
the End View of the base and intersections of the sides across it, laying off half 
the length of the base on each side of e'f, and from the extremities of the base 
drawing lines of the g-iven inclination for the ends. The interior can be dotted 
or the view shown in Section, Avhichever appears most desirable. Frequently 
the details of a structure are such that, to insure clearness, it is necessary to 
make both an external view and a section on the same plane of projection, the 
views C and D showing their arrangement in the present instance, although in 
this Figure there is no necessity whatever for both. 

N'ow complete the Plan, A, by projecting to it the external and internal 
intersections of the sides from the End View, and on these projecting lines 



MECHANICAL DRAWING. 15 

laying oiF the lengths of the intersections as obtained from the auxiliary Front 
View, C or D (remembering that any point is always at the same perpendicular 
distance from ef that it is from e'f), and drawing the diagonal lines for the 
intersections of the ends with the sides. 

We now have all the data for the completion of the Front and Side Elevations 
E and F, originally required, for which purpose draw a line perpendicular to ab 
for the horizontal base of the structure, and from this base lay off on ab the external 
and internal heights taken from view, C, and draw indefinite lines for these 
heights. Continue these lines for the heights in the Side Elevation. Project 
all the points down from the Plan to the Front Elevation. In the Side Eleva- 
tion draw the line c'd' for the trace of the vertical plane cd, and locate all 
points at the same perpendicular distance from c'd' as they are from cd. 

In case a Vertical Section on plane ab were desirable, an additional Side 
Elevation could be drawn in the position shown at G, with c"d" as the trace 
of the plane cd. 

This Figure shows how a number of views can be projected one from the 
other so that, no matter how complicated or oblique a structure may be, all 
of its parts may be clearly and definitely shown in true porportions and proper 
relation, the general principle being that all the planes of projection in a series must 
be perpendicular to each other, and that a new series of planes can be started by a plane 
perpendicular to either of the planes of the first series and oblique to all the rest. 



16 INTERMEDIATE COURSE. 

Fig. 60. — To draw a model of a corner closet, 3" high, with two sides perpendicular 
to each other, and If" wide, and a third side forming the hypothenuse of the right-angled 
triangle and having a central opening IJ" high and 1" wide, all the sides and ends 
being \" thick. 

This Figure is designed entirely for an exercise in projection. It is really 
an example of bad drawing, because more than half of the views are superfluous, 
and one of the requisites of a good drawing is that it should contain no views but 
what are necessary to give useful information. 

Draw the Plan, A, with one of its square sides perpendicular to the Front 
Elevation, and from it project 'an Elevation, B, on a plane parallel with the 
inclined side. Project a Front Elevation, C, and. a Side Elevation, D, as shown, 
making the latter a Section on the vertical plane ab. From the Plan project a 
central vertical Section, JS, on a plane perpendicular to the inclined side, and 
from ^ project an oblique rear view, F, on a plane perpendicular to the plane 
of projection of JS. From the Elevation, B, project an oblique side view, G, on 
a plane perpendicular to that of B. 



Each Figure should be inked as soon as completed, and according to the 
directions already given — that is, the fine black lines and dotted lines first, and 
then the heavy black lines, which will complete the lines of the object. Then 



MECHANICAL DRAWING. 17 

ink the blue centre lines, ab, cd, c'd', c"d", ef, e'f, and gh. Then inlc the red 
dimension lines and put in the dimensions, points and letters in black. 

Using Plate 9 as a guide, the student should devise other solids and draw- 
views of them from various points, in order to become familiar and expert in 
treating oblique views and in obtaining projections parallel to any desired part. 



DEVELOPMENTS— HEXAGONAL PRISM AND 

CYLINDER. 

PLATE 10. 

Fig. 61. — To drav) the Development of the entire surface of a Hexagonal Prism, 
2" diameter and 2^" high, cut off at an angle of 45° as shown. 

First draw the projections of the Prism, and, if any difficulty is experienced, 
refer to Plate 7, Fig. 52, of the Junior Course, which is similar. 

By the Development of the surface is meant unfolding it into one plane. All 
the sides and ends of this truncated prism are plane surfaces, but each is at an 
angle with the others, and the requirement is to draw them all in one plane in 



9B^ 



Vr* +-.c-iiiW»BWii*ee*-»^ 



18 ■ INTERMEDIATE COURSE. 

their proper relative positions so that their outline could be cut out from the 
paper or other material on which they were drawn and the surfaces then be 
folded lip to form the prism. 

The top view shows the true width of the six vertical sides of the hexagon, 
thei'efore, take this width in the spacing dividers and step it off six times on a 
horizontal line, and from each of these points erect a perpendicular to this line. 
Imagine the surface cut on the line ab, the intersection of the two largest sides, 
and unfolded from that line, then ah will be the length of the extreme vertical 
lines a'b' of the development. The line cd will be the length of the intersections 
of the front and rear sides with these largest sides; and if we make c'd' and c"d" 
equal to cd and draw b'd' and b"d", we will have these largest sides represented 
in their true size. In like manner, lay off the length efon the next vertical lines 
of the development as e'f and e"f", and gh on the central one as g'h', and con- 
nect d'f^fh', h'f" and f'd", to complete the development of the sides." Draw 
the hexagonal base, J., exactly as shown in the Plan, and copy the inclined top, 
B, from the oblique view. Every face of the solid is now drawn in its true size 
and in such relation to the adjoining faces that, if the figure be cut out from the 
paper and folded on the remaining lines, it would produce the exact form required. 

This development should be repeated on card board, the outlines cut out, 
the other intersections cut partially to facilitate folding and the object be 
produced. 



MECHANICAL DRAWING. 19 

Fig. 62, To draw the projections and development of a Cylinder, whose base is 
perpendicular to the axis and ichose top is inclined. 

Let the Cylinder be 2" diameter and the top be inclined at an angle of 45° 
with the axis and let the extreme height of the Cylinder be 2|". 

A Cylinder has a curved surface generated by moving a right line so as to 
touch a curve, all the positions of this right line being parallel. The right line 
is called the generatrix and the curve the directrix. The latter may be a curve of 
any form, but it is only necessary to consider the case of Cylinders generated 
by moving a right line in contact with a circle, all the positions of the line being 
parallel to each other and perpendicular to the plane of the circle. If another 
right line be passed through tlie centre of the circle perpendicular to its plane 
it will be the axis of the Cylinder, and will be parallel to and equidistant from 
the generatrix in all its positions. The surface of the Cylinder may thus be 
considered as made up of an infinite number of right lines equidistant from and 
parallel to the axis. Again, the Cylinder maj^ be generated by moving the circle, 
as a generatrix, along the right line as a directrix, always keeping the circle 
parallel with its first position, in which case the surface of the Cylinder may be 
considered as made up of an infinite number of circles the planes of which are 
perpendicular to the axis. Therefore, any plane perpendicular to the axis will 
cut the Cylinder in a circle; any plane inclined to the axis will cut it in an 



20 INTERMEDIATE COURSE. 

ellipse, and any plane parallel to the axis will cut it right lines. Any point on 
the surface of a Cylinder can be definitely determined by passing through the 
point two planes, one perpendicular and the other parallel to the axis, for it will 
he at the intersection of the circle produced by the first plane with the right line 
produced by the second plane. A knowledge of these simple principles will 
enable any required projections of a Cylinder to be drawn. 

In the problem under consideration, the Cylinder is 1" diameter; hence, 
describe a 2" circle for the ioi^ view, through the centre of which draw a vertical 
line ab and a horizontal line cd for the traces of two vertical planes, one perpen- 
dicular and the other parallel to the front vertical plane of projection, the 
intersection of Avhich will be the axis of the Cylinder. Draw a horizontal line 
in the Front View for the base of the Cylinder and perpendicular projecting lines 
tangent to the circle in the plan to intersect this base. On one of these projecting 
lines lay off the given height, 2J", and through this point, at the given angle, 
45°, draw the line of the top to intersect the other projecting line. This will 
complete the front view. Those portions of the projecting lines included between 
the horizontal base and the inclined top are the only visible lines produced by 
the curved surface alone. They are the lines in which the Cylinder would be 
touched by planes tangent to it and perpendicular to the plane of projection. 
Hence, the projection of a cylindrical surface consists of the traces of planes 
perpendicular to the planes of projection and tangent to the sui'face. 



MECHANICAL DRAWING. 21 

To proceed with the side view, draw the vertical line c'd' for the trace of 
the central vertical plane cd and across it project the base, on which lay off the 
diameter, 1" . At the extremities of this diameter draw indefinite vertical lines 
for the projection of the curved surface. 

As the top is inclined, its projection in the side view will be a curve, and, in 
order to draw this curve it will be necessary to find a sufficient number of points 
through which it passes by first determining these points in the top and front- 
views and then locating them in the side-view. To do this, divide the circle in 
the top-view into any number of equal parts and through each of these points 
draw vertical projecting lines to the front-view and also horizontal projecting 
lines. These projecting lines are the traces of two sets of vertical planes inter- 
secting each other in the surface of the Cylinder. Draw them also on the side- 
view by laying off from c'd' their distances from cd. Then, from each point where 
a plane intersects the projection of the inclined top in the front-view, draw a 
horizontal projecting line to the side-view and where this intex'sects the correspond- 
ing plane in that view will be the corresponding point of the curve of the top. 
Through all the points thus found draw a curve for the side-view of the top, 
which, in this case, is a circle. This will complete the side-view. 

In order to find the true shape and size of the top it is necessary to project 
it upon a plane parallel with it. For this purpose draw, parallel with the front- 
view of the top, the trace c^t/" of the central vertical plane cd., and also the traces 



22 INTERMEDIATE COURSE. 

of the other vertical planes at the same distance from c"d" as they are from cd. 
Then draw projecting lines from all the points in the front- view, which have 
already been determined, to this auxiliary -view; and, where these lines intersect 
the corresponding vertical planes will be the positions of the points in this view. 

The similarity between this Figure and the Hexagonal Prism in Fig. 61 
should be noted, because the curved surface of the cylinder has been divided by 
lines into a number of spaces, and, as regards the lengths of these lines, the 
treatment is the same as if the surface was made up of that number of plane 
faces. These lines on the surface of the Cylinder need not be considered as the 
traces of planes intersecting it, but may be merely as lines at regular distances 
apart around the circumference. Then, if these lines are properly drawn in all 
the views, and a point be determined on one of the lines in one of the views, the 
same point can be readily found in all the other views. 

To draw the development of the entire surface of this Cylinder, proceed as 
with hexagonal prism in Fig. 61. Draw a horizontal line, and on it lay oft' the 
length of the circumference of the Cylinder as found by calculation. Divide 
this length into the same number of equal parts as those in the circle in the io'p 
view. At each of these divisions erect a perpendicular, and on each perpen- 
dicular lay off" the height as obtained from the corresponding line in the front 
view. A curve passed through these points will complete the development of 
the cylindrical surface. The base will be a 2" circle the same as in the top view, 
and the top will be an ellipse the same as in the auxiliary view. 



MECHANICAL DRAWING. 23 

PYRAMIDS. 
PLATE II. 

Fig. 63. To draw the development of a 2" Hexagonal Pyramid, 2i" high. 

Draw the Pyramid as in Plate 7, Fig. 51, of the Junior Course. 

Each side of this Pyramid is a triangle, the apex of which coincides with 
the apex of the Pyramid; hence, select a convenient point for a centre, and, 
with radius equal to the length of a side of the triangle, describe a portion of a 
circle, and on this step off chords the length of the base of the triangle as many 
times as there are triangular sides to the Pyramid, making sure to obtain the 
true length of both the side and the base of the triangle. Connect each of these 
points by right lines with the centre of the circle and with each other for the 
development of the inclined surfaces, and draw the hexagon of the base adjacent 
to the base of either ,of the triangles. 



Fig. 64. To draw the development of a 2" Hexagonal Pyramid 2^" high, cut off 
by a plane parallel to and %" from the axis, the plane being perpendicular to one of the 
sides of the base. 

First draw the projections of the entire Pyramid and its complete develop- 
ment, as in Fig. 63. Then cut it by the given plane and draw the projections 



24 INTERMEDIATE COURSE. 

of the cut. From the front view take the length of the line ca, which is left by 
the cut, and lay this length off on the corresponding line in the development. 
The line ch is not shown in its true length in the front view, because it is not 
parallel with the plane of projection ; but by drawing a horizontal line from the 
point 6 to a line of the Pyramid which is parallel to the plane of projection, 
the true length cb' will be obtained. Cut off the development of the base the 
same as in the j)lan, and transfer the vertical cut surface from the side view to the 
development, being careful to bring either of its edges in contact with the proper 
edge of the rest of the development. 

As an exercise, other Prisms and Pyramids, cut in different places, should be 
assumed, their projections drawn, and their developments cut out of cardboard 
and folded up to produce the forms. 

THE CONE. 

PLATE 12. 

Pig. 65. To draw the projections of a Cone 2^" high and 2^" diameter at the 
base, cut off by a plane parallel with and \" from the axis, and to draw the development 
of the entire surface. 

A Cone has a curved surface generated by moving a right line so as to 
touch a curve and at the same time to pass through a fixed point not in the plane 



MECHANICAL DRAWING. 25 

of the curve. The right line is the generatrix, the curve the directrix. Those 
cones only which have a circular directrix, and in which the generatrix passes 
through a point in a line perpendicular to the centre of the plane of the directrix, 
will be considered. This perpendicular line is the axis, the point the apex, and 
the cone a right cone. 

It is evident that no right line can be drawn on the conical surface "without 
passing through the apex, and that any plane which passes through the apex will 
cut the surface in right lines, if at all ; also that any plane perpendicular to the 
axis wall cut the surface in a circle the radius of which will be the perpendicular 
distance of any point of the circle from the axis. Hence, if it is required to 
determine any point on the surface, it is only necessary to pass two planes 
through this point, one being perpendicular to the axis and the other containing 
the apex. 

The Cone under consideration has a base 2^" diameter. Locate the centre 
of the base in the top-view and through this centre draw the verticul centre-line 
ab and the horizontal centre-line cd, the former of which will be the trace of a 
vertical plane perpendicular to the front plane of projection and the latter the 
trace of a vertical plane parallel to the front plane of projection. The intersection 
of the planes ab and cd will be the axis of the cone. About this axis, describe a 
circle, 2^" diameter. Continue ab as a'b', which will be the trace of the same 
vertical plane in the front^vuw. On a'b' lay out off the height of the cone, 2J", 



26 INTERMEDIATE COURSE. 

and through the lower point thus marked off draw a horizontal line for the base. 
Project the extremities of the diameter of the base from the top-vieio to this line, 
and draw inclined lines from these points to intersect in the apex already laid 
oif on the axis. At a convenient distance from a'b' draw the vertical line c'd', 
which will be the trace of the plane cd on the side vertical plane of projection. 
From the front-view project the apex and the line of the base to the side-view and 
on this line lay off the diameter of the base by marking off the radius on each side 
of the plane c'd' which is an elevation of the plane, cd. Then draw the inclined 
lines from the extremities of the base to the apex, the same as in the front-view. 
Having now three views of the entire Cone, it is necessary to draw the plane 
which is to cut it, parallel with the axis and f " from it, according to the proposi- 
tion. Let this cutting-plane be perpendicular to the front. Its trace will be a 
line parallel with and f" from the axis in both top and front views, but its actual 
line of intersection with the conical surface will be curved and not straight. The 
true shape of this curve of intersection will appear in the side-view, and must be 
determined from the straight lines which constitute its projections in the tojo and 
front views. Fix upon any number of points at any distances apart on the line 
\n the front-view and through them pass horizontal planes. The projections in 
the top-view of the intersections of these planes with the conical surface will be 
circles, the radius of each of which will be equal to the distance from the axis to 
the point wliere the plane cuts the conical surface. In the top-view, describe these 



MECHANICAL DRAWING. 27 

circles so as to cut the vertical cutting-plane. Draw the traces of these same 
horizontal planes in the side-view and on them lay off from the central vertical 
plane c'd' the distances from the same plane cd in the top-vieio of the points where 
the circular traces of the corresponding horizontal planes intersect the cutting- 
plane. A curve passed through the points thus found will be the true shape of 
the line of intersection of the vertical cutting-plane with the conical surface. 
This curve is a hyperbola. 

The problem thus far could also have been solved by drawing in the three 
views, the traces upon the conical surface of a series of vertical planes passing 
through the axis and projecting the points of their intersection with the cutting- 
plane from the /ron^i/feiu to the corresponding traces in the side-view; but this 
method is not as accurate in this instance on account of the acute angle at which 
the traces intersect the plane. These traces will assist materially, however, in 
drawing the development of the surfaces of the Cone. 

To construct this development, divide the circle of the base in the top-view into 
any number of equal parts, and draw diameters through these points. These 
diameters will be the traces of a series of vertical planes passing through the axis 
of the Cone. Project the extremities of these diameters to the front and side- 
views of the base, and draw lines from these points on the base to the apex. 
These lines will be the traces upon the conical surface of the same series of 
vertical planes, and, as the Cone is symmetrical about the axis, each line will be 



28 INTERMEDIATE COURSE. 

of the same length and will have the same inclination to the axis as the side of 
the cone. Hence, describe an arc of a circle, with radius equal to the length of 
the side, and step oflt'on this arc divisions of the same number and length of arc 
as those already made in the circle of the top-view, and then connect these 
divisions with the centre, and the result will be the development of the entire 
conical surface. 

To cut awaj the same portion of the development as has already been done 
of the elevations, it is only necessary to determine in the former the location 
of the points which have already been fixed in the latter. These points are the 
intersections of the traces of a series of horizontal planes with the vertical cutting 
plane. Draw the traces of the horizontal planes in the development by describing 
arcs of radii equal to the distances from the apex to the traces on the conical 
surface, as shown in the front vieu:. In order to locate the required points on 
these arcs, determine their positions in relation to the central plane cd in the 
top view by drawing lines through them from the apex to the circle of the base. 
Draw these same lines in the development by transferring their points from the 
circle in the toj) vieio to the arc in the development. Where the lines intersect the 
corresponding arcs in the development will be the points required, and curves 
drawn through the points will cut away the development to correspond with the 
elevations. Copy the base of the cone from the top view, and, adjoining it, copy 
the hyperbola from the side vieiv to complete the development, which, if cut out 



MECHANICAL DRAWING. 29 

from the paper and properly rolled and folded, will produce the solid shown in 
the drawing. 



Fig. 66. To draw the 'projections of a Cone 2J" high and 21" diameter at the 
base, cut off by a plane which makes an angle of 60° with the axis and which cuts the 
surface at a perpendicular distance of 1 inch below the apex, and to draw the develop- 
ment of the entire surface. 

Draw the three views of the entire cone, as in Fig. 65, and, in i\\Q front view, 
draw the trace of the cutting plane as given. Draw an auxiliary view on a plane 
parallel with the cutting plane. 

It is first necessary to determine points on the line of intersection of the 
cutting plane with the conical surface in the front view, which can he done by 
intersecting it Avith a series of eitlier horizontal or vertical planes, whichever 
can be most readily used or produce the most accurate results. It is evident 
that horizontal planes will intersect it at much more acute angles than 
vertical planes, and that the insersections will be less distinctly defined; 
hence it is best to use vertical planes, which, as has already been shown, must 
all pass through the apex of the cone. Hence, in the top-view, divide the circle 
of the base into any number of equal parts and draw diameters through these 
divi^'ions for the traces of a series of vertical planes passing through the apex, 



30 INTERMEDIATE COURSE, 

Draw the traces of these planes upon the cone in the three other views. From 
the points in the front-view where these traces intersect the trace of the cutting 
plane, draw projecting lines to intersect the traces in the other views. Curves 
drawn through the latter intersections will give the projections of the cut surface, 
the true size and shape of which will be given in the auxiliary-view, which is a 
projection on a plane parallel with this surface. The shape is an Ellipse, as is 
always the case when the cutting plane passes through both sides of the cone. 

The surface is developed by describing an arc of a circle of I'adius equal to 
the length of the side of the cone, stepping off on this arc the same divisions as 
were used in the top view, connecting these divisions with the centre by lines, in 
order to represent upon a plane the traces already used on the conical surface, 
and laying oiF the true length of these traces, as explained in Pig. 64. A circle 
1\" diameter and an ellipse copied from the one in the auxiliary view completes 
the development, which should be cut out of the paper and put into the form of 
the solid. 



The student should draw several different Cones of diife rent relative heights, 
and should cut them by vertical planes at different distances from the axis, and 
also by inclined planes at different angles with the axis, 



MECHANICAL DRAWING. 31 



PLATE 13. 

Fig. 67. To draw the j)rojedions of a Cone 2\" high and 2^" diameter at the 
base, cut off by a pldne parallel to and f " from the side. 

Draw the three views of the entire cone as before, and, in the, front view, 
draw a line parallel to the side of the cone and. f " from it for the trace of the 
cutting plane. In this instance it will be best to use horizontal planes to 
intersect the cutting plane for the purpose of determining certain points of its 
intersection with the conical surface. Hence, draw the traces of any number of 
horizontal planes in the front and side vieios, and, in the top view, draw the 
circular traces which these planes make upon the surface of the cone. Each of 
these horizontal planes will draw a trace upon the cut surface, and the length of 
each trace is the chord of the arc of the circular trace of the same horizontal 
plane as shown in the top view. Hence, from the front view draw projecting lines 
from the points of intersection of each horizontal plane, with the cutting plane 
to intersect the circular traces in the top view, and these intersections will deter- 
mine the points in relation to the central plane cd. On the horizontal traces in 
the side view lay off from c'd' on each trace the distances from cd in the top view 
of the points in the corresponding trace. Curves drawn through these points 
will complete the top and side views. 



32 INTERMEDIATE COURSE. 

To obtain the true size and shape of the cut surface, the cone should be 
projected upon a plane parallel with this surface. To do this, draw c"d" parallel 
to the trace of the cutting plane in the front view, for the trace in an auxiliary 
view of the central plane cd in the top vino, and then draw the projections of the 
base and apex of the cone in this auxiliary view, the former of which will be an 
ellipse, and the latter a point. Prom the apex draw tangents to the ellipse to 
complete this view of the entire cone. Then project upon this view the traces 
of the horizontal planes upon the cut surface, and lay off" the lengths of these 
traces as obtained from either the iajp or side view. A curve drawn through 
these points will give the exact shape and size of the cut surface. 

This curve is a Parabola, as are all curves formed by the intersection of a 
right cone of any proportions with a plane parallel to its side. 



Fifl. 68. To draw the projections of a Tube 2\" outside and If" inside diameter, 
cut off' by a 2^ lane at an angle of 60° rvith the side, from a point 2^" from one end. 

This problem is similar to the one in Fig. 62, but is to be solved by a 
more rapid method, which, although not as accurate, is still sufficiently so for 
most practical purposes. 

Draw the trace fl6of the central vertical plane, and, in a location convenient 
for the top view, draw the trace cd of the other centi-al vertical plane, perpen- 
dicular to the first. The intersection of these planes will be the trace of the axis 



MECHANICAL DRAWING. 33 

of the Tube in the top-view. About this point as a centre, describe a circle, 2^" 
diameter, for the exterior of the Tube, and one, 1|" diameter, for the interior. 
Draw a horizontal line for the front-view of the lower end, set the triangle tangent 
to the circles in the top-view, and draw indefinite lines up from this lower end 
for the front elevation of the Tube, the top being as yet undetermined. Lay off 
the given length 2J", and through this point draw the trace of the cutting plane 
at the given angle, 60°. 

DraAV, in the side-view, the trace c'd' of the central vertical plane cd and the 
base and sides of the Tube, leaving the top indefinite. As the line of intersec- 
tion of a Cylinder by a plane, which cuts the axis, is either a circle or an ellipse, 
the side-vieiv of the top in this case will be two ellipses, and it is frequently 
allowable to approximate these ellipses by means of circular arcs, a method of 
finding the centres of which is given in Fig. 32, Plate 3, of the Junior Course. 

From the point of intersection of the cutting-plane and the axis in the front- 
view, draw a horizontal projecting line across the side-view of the Tube. This 
will contain the major axes of the ellipses. Also, from the front-view, project 
the points where the trace of the cutting-plane intersects the exterior and interior 
of the Tube, to the line c'd' in the side-view. These points will determine the 
minor axes. On these axes construct the approximate ellipses with circular arcs. 

To obtain an auxiliary -view parallel with the cutting plane, draw, parallel 
with this plane, the trace c"d" of the central vertical pli^ne cd, and upon this 



34 INTERMEDIATE COURSE. 

line project from the front view the external and internal extremities and the 
centres of the top and base. These centre lines will limit the parallel lines of 
the tube in this view ; therefore lay off on one of them the diameters of the tube 
and draw the lines parallel with c"d" . All the points necessary for drawing 
approximate ellipses for the top and base in this view are now obtained. 



Fig. 69. To intersect a cone of two nappes by a plane forming an angle with the 
axis and cutting both nappes. 

By a cone of two nappes is meant one Avhich is produced by a generatrix 
which continues beyond the apex, the result being two equal and similar 
cones having the same axis and the same apex, but tapering in opposite 
directions. 

Let each cone be 4J" high and 3" diameter at the base. Draw the top and 
front views as shown. Di-aw the trace of the cutting plane in the front view so 
that it intersects the lower base If" from one extremity and the upper base I" 
from the opposite extremity. In the top view draw the lines of intersection of 
the cutting plane with the upper and lower bases. Draw the traces of a series 
of horizontal planes in \\\& front and top views. Project the points of intersection 
of the cutting plane with these traces in the front view to the corresponding 
traces in the top view, and draw the curves through the points thus obtained in 



MECHANICAL DKAWING. 35 

the top view. These two views give all necessary information, excepting the 
true form of the cut surfaces. These will be Hyperbolas, as is always the case 
when a cone is intersected by a plane which does pass through both sides and 
is not parallel with either. 

To find the form of these surfaces, a projection must be made on a plane 
parallel with the cutting plane by making the trace c'd' of the vertical central 
plane cd parallel with this cutting plane, and about c'd' , which, of course, con- 
tains the axis, completing the auxiliary vieio of the entire cone. Then upon this 
view project from the front view the traces of the intersections of the horizontal 
planes with the cutting plane (the lengths of these traces being obtained from 
the top view), and draw the resulting hyperbolas. 

A curious fact will then become apparent, namely, that the hyperbola 
produced upon the upper nappe is precisely the same curve in form and size 
as the one on the lower nappe, and this will ahvays occur when the same plane 
cuts both nappes. 



36 INTERMEDIATE COURSE. 

INTERSECTIONS OF SOLIDS HAVING PLANE 

SURFACES. 



PLATE 14. 

Fig. 70. To draw a vertical prism If" square and 3" long, intersected by a hori- 
zontal prism IJ" square and 4" long, one side of each prism to make an angle of 45° 
loith the front plane of projection, and the axes to intersect at their centres. 

The importance of trainirig the miiid so as to be capable of a clear concep- 
tion of the intersections of solids, will be appreciated when the fact is considered 
that all structures contain such intersections, which may be, and often are, the 
only part of the structure where any difficulty is experienced in the designing 
or drawing. The principles involved are really simple, and the solution of 
apparently complicated and difficult cases becomes easy after the habit is acquired 
of reducing each case into its elements. 

To proceed with the present case, locate the traces of the central vertical 
planes ab and cd in the front, top and side views, and draw the If" square end 



MECHANICAL DRAWING, 37 

of the vertical prism in the top view. Lay off the height 3" in the front view, 
and, midway of this height, draw the trace ef of a central horizontal plane in 
both front and side views. At the intersection of ef and c'd' draw the IJ'' 
square end of the horizontal prism, and lay off the length of the latter, 4",. in 
the front view. 

Draw the ends of the vertical prism in the front and side views and project 
its corners to them from the top view. Now, as the diagonal of the square of this 
prism is longer than that of the horizontal prism, the front and back corners 
of the former will not touch the latter; hence the lines of these corners will 
extend uninterruptedly from the top to the bottom; therefore draw these lines 
complete in the front and side views. 

The right and left-hand corners of the vertical prism being in the same 
plane with the top and bottom corners of the horizontal prism, these corners 
will intersect, and their points of intersection are immediately obtained from 
the side view, as at x, the front view of which is at x' and the top view at x". 
Complete the projections of these corners in all the views. 

As the diagonal of the square of the horizontal prism is shorter than that of 
the vertical, the front and back edges of the former prism will intersect the four 
sides of the latter in points as yet undetermined. To find these points, project 
the ends of the horizontal prism from the front to the top view, lay otFthe length 



38 INTBBMEDIATE COURSE. 

of the diagonal on the latter, draw the corners until they intersect the sides of 
the vertical prism, and project these intersections y to the front-view, as y' . 

The points of intersection of all the corners of the horizontal prism are now 
determined in all the views, but, as the sides of both prisms are inclined to the 
front plane of projection, it is evident that the intersection of these sides will 
show in the front-view. These intersections must be right Unes, because the 
sides are planes, and the intersections of planes can only be right lines. As the 
lines of the corners are included in the planes of the sides, and as the points 
of intersections of the corners are already determined, the intersections of the 
sides must be right lines joining these points, as x'y' . 



Fig. 71. To draw a vertical prism If" square and 3" long, with its side at an 
angle of 15° loiih the front plane of projection, intersected by a horizontal prism IJ" 
square and 4" long, with its axis parallel to and its side at an angle of 15° ivith the front 
plane of projection, the axis of the horizontal prism passing \" in front of that of the 
vertical prism. 

Draw the traces ab, cd, c'd', efgh and g'h' of the central, vertical and horizontal 
planes which contain the axes of the two prisms, cd being ^" in front of gh. About 
the intersection of ab and gh in the top view, describe the If" square end of the 



MECHANICAL DRAWING. 39 

vertical prism, with its side at the given angle of 15°, and about the intersection 
of €/ and ^'A', in the side-view, draw the IJ" square end of the horizontal prism 
with its side at 15°. 

Complete the projection of the horizontal prism in the top-view and that of 
the vertical prism in the side-vieM^ Then the top-view will immediately give the 
points where the corners of the horizontal prism intersect the sides of the vertical 
prism, and the side-view those where the corners of the vertical intersect the 
sides of the horizontal. Take for instance the front right-hand corner of the 
vertical and the front upper corner of the horizontal prisms. The former inter- 
sects at point X in the side-view, the other projections of which are x' in the 
front- view and x" in the top-view. The latter intersects at points y and z in the 
top-view, y' and z' in the front-view and y" in the side-view. Lines connecting 
x' with y' and z', will give the lines of intersections of these sides in the front- 
view. Find the points of intersection of the other corners with the other sides 
in all the views and the lines of intersection of the sides in the front-view. 

An analysis of this figure will show that in dra\^ing it the following problems 
have been solved : — the intersection of vertical lines with planes perpendicular to 
one vertical plane of projection, but inclined to the other; — the intersection of 
horizontal lines with vertical planes inclined to the vertical planes of projection ; 
— and the intersection of planes M'hich are perpendicular to one but inclined to 
the other two planes of projection. 



40 INTERMEDIATE COURSE. 

Fig. 72. To draw a vertical hexagonal prism 2" diameter and 3|" long with one 
side parallel to the front plane of projection, intersected by an inclined prism 1\" square 
and, 4" long, the axes intersecting at an angle of 45° at a point in the centre of thejormer 
and 1^" from the upper end of the latter, the sides of the latter being at 45° with the 
front plane of projection. 

Draw the traces ab, cd and c'd' of the central vertical planes, lay off on ah 
the height, 3|", of the hexagonal prism, draw indefinite lines for its ends, and 
mark the half. If", of this height, through which, at an angle of 45°, draw the 
trace e/of a plane perpendicular to the front plane of projection and containing 
the axis of the inclined square prism. On ef, from its point of intersection with 
ah, lay off If" upwards and 2J" downwards for the length, 4", of the inclined 
prism and draw indefinite lines for its ends. In the top-view, ahout the inter- 
section of the central vertical planes ab and cd, describe the 2" hexagonal end of 
the vertical prism. To draw the square end of the inclined prism, select a point 
on e/" through which to pass a trace c"d" of the central vertical plane cd upon a 
plane parallel to the end of the inclined prism. Then the plane c"d" must be 
perpendicular to ef, because e/is perpendicular to the front plane of projection, 
to which cd is parallel. Ahout the intersection of c"d" and ef, describe the 1\" . 
square end of the inclined prism with its sides at the given angle 45° with c"d". 

Project the extreme right-hand edge of the vertical prism from the top-view 
to the front-view until it intersects the end of the inclined prism, and then 



MECHANICAL DRAWING. 41 

project this point of intersection to the side-view and the auxiliary view, drawing 
this much of the edge in these views. Project the lower edge of the square 
prism from the auxiliary view to the front view where it will meet the right- 
hand edge of the vertical prism, and will determine their point of intersection, 
which point is then to be projected to the side-view and the edge completed. 
The point where the lower edge of the inclined prism emerges from the base of 
the vertical prism, is determined in the front-view and projected from this to the 
top-view. The front and back edges of the inclined prism do not touch the 
vertical prism, but show continuous lines in the fioiit, top and side views, while 
the upper edge intersects as shown in the front-view, from which the points are 
projected to the top and side views. The upper square end of the inclined prism 
intersects two sides and the top of the vertical prism, the former points being 
obtained from the top-view, and the latter points from the front-view. 

The front and back sides of the vertical prism intersect the sides of the 
inclined prism in lines determined by the auxiliary view, and from which the 
projections ai'e made. 

So far, every point of intersection has been directly obtained from either 
one or the other of the views, and this can always be done with objects whose 
lines are parallel with the central planes. If one branch of the object is inclined, 
it is only necessary to make an auxiliary view on a plane perpendicular to the 
central plane of the inclined branch in order to get the points of intersection 



42 INTERMEDIATE COURSE. 

with it. Although this is not geometrically necessary, it is more accurate, and 
the view thus obtained is very useful in making the drawing more clear and 
complete, and is often necessary to enable a correct construction of the object to 
be made. 

In order to show that the points can be obtained without the auxiliary view, . 
the base of the vertical prism in the latter has not been completed; but its 
intersection with the lower sides of the inclined prism have been determined in 
the following manner; — 

If the plane of the base of the vertical prism were continued, it would 
intersect the lower end of the inclined prism in a line of which the point x is 
the projection in the front view and the linex'x" in the top view, and it intersects 
the edge of the inclined prism in a point y in the front view of which y' is the 
projection in the top view. Hence, y'x'x" shows the projection in the top 
view of the intersection of the plane of the base of the vertical prism with the 
inclined prism, and the points where these lines cut the hexagon are the ones 
required. 

In further illustration of this, let it be required to find the intersection of 
the front side of the vertical prism with the side of the inclined ])ri8ra without 
the use of the auxiliary view. Continuing the plane of the former in the top 
view until it intersects the ends of the latter, project these points of intersection 



MECHANICAL DRAWING. 43 

to the front view and draw a line connecting them, then as much of this line 
as is contained within the former will be the front view of the required 
intersection. 

It is-alwajs possible, and sometimes convenient, to find the intersection of 
inclined sides of objects by using only two views; but as the object of drawing 
is to make the construction clear, and not merely to display knowledge of 
geometr}', the former should not be sacrificed to the latter. 

In Fig. 72, the top and front views fully determine, in a geometrical sense, 
every point of the object, but a good mechanic would be sorely puzzled in 
attempting to construct it from these views alone, unless he were told that the 
inclined prism was to be 1|" square. The side view, however, could be 
dispensed with, as it conveys no additional information. 



Fig. 73. To draw a vertical hexagonal prism 2" diameter and SI" long with one 
side parallel to the front plane of projection, intersected on one side only by another 
hexagonal prism li" diameter at an angle of 30° with the horizontal plane, the front 
side of both prisms to be in the sam.e vertical plane, the planes of the axes to intersect at 
a distance of IJ" above the base of the vertical prism and the end of the inclined, prism 
to be at a distance of 2^" from this intersection. 



44 INTERMEDIATE COURSE. 

Draw the traces of the central vertical planes ab, cd and c'd', and about 
these construct the top, front and side views of the vertical prism. In the front 
view lay off on ab a point IJ" above the base, and through this, at an angle of 
60° with ab, draw ef for the trace of an oblique plane containing the axis of the 
inclined prism. Perpendicular to this, draw c"d" for the trace in an auxiliary 
view of the vertical plane cd, and complete this view of the vertical prism. 

As the difference of the diameters of the two prisms is J", and as their 
front sidesare in the same vertical plane, the vertical planes of their axes will 
be \" apart. Therefore, draw the traces gh, g'h' and g"h" in the top, side and 
auxiliary views of the central vertical plane of the inclined prism. In the 
auxiliary view the intersection of g"h" and ef will be the axis of the inclined 
prism, about which draw the 1\" hexagon. On ef, lay off 2J" from ab, and at 
this point draw a perpendicular to ef for the plane of the end of the inclined 
prism in the front view. 

The points of intersection of the edges of the inclined prism with the 
vertical prism can be obtained directly from the top view. Take, for instance, 
the bottom edge x y. It intersects at point y in the top view, which is projected 
down to y' in front view and then across to y" in side view; and so with all the 
other edges. The intersections of the edges of the vertical prism with the inclined 
prism are obtained from the auxiliary view. 



MECHANICAL DRAWING. 45 

It will be noted that in this Figure the views which are iiseful for con- 
structive purposes are Top, Front and Auxilliary Views, while the Side View 
could be dispensed with, excepting that in actual practice it is generally useful 
and often important for other purposes. 

PLATE 15. 

Fig. 74. To draiv a vertical Pyramid S" high, having a triangular base 5" long 
on each side, with the rear side at an angle of 15° with the front plane of projection, the 
Pyramid intersecting a horizontal Prism %" long, having triangular ends 4J'' long on 
each side, with the rear side parallel with the front plane and at a distance of 1^" back 
of the axis of the Pyramid, the horizontal centre line of the Prism to be SJ" above the 
base of the Pyramid. 

Draw the three views of the vertical pyramid, commencing with the plan 
of its base in the top view. Also draw the three views of the horizontal prism, 
commencing with its end in the side view. • The points of intersection can then 
be obtained directly from the side view, projected across to the front view and 
up to the top view. 

It has been already stated that two views are all that are necessary to 
determine all the points of a solid ; and, although three views are generally 
more convenient and desirable, yet sometimes it becomes difficult and tedious 



46 INTERMEDIATE COURSE. 

to employ all the views necessary to obtain directly all the points of intersection. 
As this Figure is a good example of the intersection of inclined lines with inclined 
planes, it will he well to analyze it for the purpose of understanding how such 
points of intersection can be obtained from the top and front views only, with- 
out the use of planes of projection perpendicular to the inclined planes. 

Let it be required to find where the inclined line forming the front edge 
of the vertical pyramid intersects the two inclined faces of the horizontal prisnci. 
It is evident that, if we intersect these inclined planes by a vertical plane con- 
taining this inclined line, the vertical plane will draw traces on the inclined 
planes and that these traces will contain the points of intersection, because the 
line is contained by the plane which makes them. 

In the Figure, the trace of this vertical plane in the top view is ab, which, 
projected to the front view, gives a'b' and a'b" as the traces on the inclined 
faces. These traces intersect the inclined line of the front edge of the pyramid 
at X and z', which are the points required. 

In the same manner for the left-hand edge, draw the trace ed of its vertical 
plane in the top view, and project this to c'd' and c'd" in the front view to 
obtain the points ?/ and y'. 

For the right-hand edge, draw the trace no in the top view, extending the 
limiting lines of the inclined faces far enough to obtain the intersections, and 
project to n'o' and n'o" in the front view to obtain the points z and z'. 



MECHANICAL DRAWING. 47 



PLATE i6. 

Fig. 75. To draw the same Pyramid and Prism as in Fig. 74, the Pyramid 
being in a similar position, but the Prism being inclined at an angle of 60° with the 
perpendicular, its central plane crossing that of the Pyramid at a height of 4J inches, 
and its vertical side being 2^ inches back of the axis of the Pyramid. 

In this case it is doubtful whicli of the two methods explained with Fig. 74 
is the quickest and most reliable. If it is desired to find the intersections by 
projection only, a complete auxiliary view, projected upon a plane parallel with 
the end of the prism, will be required. This would be the clearest and most 
easily understood by a mechanic proposing to make the object ; but, for the 
purpose of gaining a better understanding of the descriptive geometry involved 
in the use of only two views, it will be best to use the latter method. 

Draw the top and front views of the Pyramid complete, and, in tlie front 
view draw the centre line of the prism crossing the axis of the pyramid 4^ inches 
above its base and at an angle of 60° with the perpendicular. On this centre line 
draw an auxiliary end view of the Prism, and from this make its top and front 
proj<^ctions complete. 



48 INTERMEDIATE COURSE. 

The problem now is similar to the one in Fig. 74 — that is, there are the 
three inclined planes of the Pyramid intersecting the two inclined planes of the 
prism, with this difference, that the latter incline not only to the front vertical 
plane, but also to the side vertical plane. 

As each of the planes of the Pyramid is limited by the corners, it is only 
necessary to find the points of intersection of these corners with the sides of the 
prism. Take the line of the front corner and imagine a vertical plane which 
contains it to be passed through the prism. Such a plane would draw the traces 
ab, a'b' and a'b" upon the sides of the prism, and the points x and x\ where these 
traces cut the line in the front view, will necessarily be the points of intersection 
required, which are then projected to x" and x'" in the top view. 

A vertical plane containing the right-hand corner will draw the traces no, 
n'o' and n'o" upon the sides of the prism, but it is necessary to imagine these 
sides extended far enough to reach the points n and o. Where these traces 
intersect the edge will give the points 2-, z' in the front, and 2", z'" in the top 
views. The points of -intersection y and y' of the left-hand edge are found in 
the same manner. 

Having found the points where the inclined lines of the edges of the vertical 
Pyramid pierce the inclined planes of the sides of the Prism, then lines xy, xz, 
yz, and x'y\ x'z\ y'z' will be the'lines of intersections of the inclined planes of 
the Pyramid with the inclined planes of the Prism. 



MECHANICAL DRAWING. 49 



PLATE 17. 

Fig. 76. To draw a vertical Pyramid 6J" high, having a triangular base 6f " 
long on each side, with the rear side at an angle of 15° with the front plane of projection, 
the Pyramid intersecting a Prism inclined at an angle of 60° with the perpendicular, 
the axis of the former piercing the central plane of the latter at a height o/3f" above the 
base, the Prism being 9" long and having triangular ends 3J" long on each side, with 
the rear side parallel with the front plane of projection and at a distance of 1^" behind 
the axis of the Pyramid. 

Draw the top and front views of the Pyramid and Prism complete. 

In attempting to find, as before, the points where the front edge of the 
Pyramid pierces the sides of the Prism by drawing the traces on the latter of a 
vertical plane containing the former, it will be discovered that these traces do 
not reach the line, and that, therefore, the line passes entirely outside the Prism. 
This being the case, it follows that the front edge of the Prism must pierce the 
sides of the Pyramid, and therefore the traces of a plane containing this front 
edge must be drawn upon these sides. Either view can be selected for this 
purpose, whichever is more convenient or accurate. In the present instance, 
draw upon the top view of the sides of the Pyramid the tfaces of a plane, per- 
pendicular to the front plane of projection and eontaining the line of the front 



50 INTERMEDIATE COURSE. 

edge of the prism. These traces, in the top view, will be a'b' 

intersect the line in the required points x and y in the top view, from which are 

projected the points x' and y' in the front view. 

It will now be understood that the method of determining the point where 
an inclined line pierces an inclined plane is often a matter of judicious selection 
or of invention, and that the best possible training for such operations is to 
assume difi'erent plane solids in various positions, and find their intersections. 
Therefore, the remainder of the intersections in this Figure are required to be 
found without farther explanation. 

PLATE i8. 

Fig. 77. To draw a vertical Pyramid 7J inches high, having a triangular base 
4 inches long on each side, with the rear side at an angle of 15° with the front plane of 
projection, the Pyramid intersecting another Pyramid 9 inches long, having also a ^-inch 
triangidar base, the axis of the second Pyramid intersecting that of the first at an angle 
of 60°, and at a point 3f inches from the base of the second and 4 J inches from the base 
of the first, the rear side of the second also making an angle of 15° with the front plane 
of projection. 

This is a case in which there would be no benefit whatever in making more 
than two views, for the reason that no two sides are parallel, and no plane of 



MECHANICAL DRAWING. 51 

projection could be selected which would be perpendicular to more than one of 
them. It is an excellent example of the penetration of inclined planes by 
inclined lines and of the intersection of inclined planes. 

Draw the top and front views of the Pyramids complete. Then a vertical plane 
containing the front edge of the vertical Pyramid will draw the traces ab and ac 
upon the sides of the inclined Pyramid, and where these traces intersect this 
edge in the front view will be the points where it pierces the sides, which points 
are then projected to the top view. 

A vertical plane containing the left-hand edge of the vertical Pyramid will 
draw the trace de upon the upper side of the inclined Pyramid, and where de 
intersects this edge in the front-view will be the point where the edge pierces 
the upper side. But if a trace of this same vertical plane be drawn on the lower 
side, it will be found not to intersect the edge ; hence, the edge must pierce the 
rear side, the trace on which is dn, which does intersect the edge, giving the 
point where it pierces the rear side. 

A vertical plane containing the right-hand edge of the vertical Pyramid will 
draw the traces^ upon the upper and hm upon the lower sides of the inclined 
Pyramid and where ^ intersects this edge in the front- view will be the point 
where the edge pierces the upper side, and where hm intersects it will be the 
point on the lower side. 



52 INTERMEDIATE COURSE. 

The points where the three edges of the vertical Pyramid pierce the sides 
of the inclined Pyramid being now determined, the next step is to connect these 
points for the lines of intersection made by the sides of the two Pyramids. The 
three points on the upper side of the inclined Pj-ramid can be readily connected; 
also, the points where the front and right-hand edges pierce the lower side; but. 
a difficulty ai'ises with the left-hand edge from the fact that it pierces the rear 
side, and a single line of intersection cannot be drawn on the surface of the 
inclined Pyramid which will connect a point on the rear side with two points on 
the lower side. The inference from this is, that the lower edge of the inclined 
Pyramid must pierce the Vertical Pyramid. 

To find these points, cut the vertical Pyramid, in the front-view, by a plane 
perpendicular to the front plane of projection and containing the lower edge of 
the inclined Pyramid. This will draw the traces op on the rear and ps on the 
left-hand sides of the vertical Pj^ramid, in the top-view, and where op cuts the 
lower edge, will be the point where it pierces the rear side, and where ps cuts it, 
will be the point where it pierces the left-hand side. Then these points can be 
connected with those previously found to complete the intersection of the two 
Pyramids. 

The study and analysis of the intersections of Solids with Plane Surfaces 
form such an excellent training for intricate and difficult problems in architectural 



MECHANICAL DRAWING. 53 

and engineering construction, that the importance of a complete mastery of the 
subject cannot be overestimated. ISTumerons combinations of solids of various 
-forms should be draw^n in various positions, and their intersections carefully 
worked out, when it will soon be found that difficulties, which at first seemed 
insurmountable, can be readily overcome. 



INTERSECTIONS OF SOLIDS HAVING CURVED 

SURFACES. 

PLATE 19, 

Fig. 78. To draw a vertical Cylinder 2J inches diameter and 2f inches long, 
intersected by two horizontal cylinders of the dimensions and in the 'positions shown, and 
to draw the development of all the surfaces. 

As explained in Plate 10, Fig. 62, any plane parallel with the axis of a 
cylinder will cut its surface in straight lines. By using a series of planes to 
trace lines on the surface of a cylinder, and finding the points where these lines 
pierce the intersecting cylinder and then connecting these points, the problem 
becomes a simple one. 

In the present instance, cut the side views of the cylinders by vertical planes 
(the more the better), draw, in the top view, the traces of these planes upon the 



54 INxiiRMBDIATE COURSE, 

surface of tlie horizontal cylinders, and project the points where these traces 
pierce the vertical cylinder to the traces of the vertical planes in the front view, 
for the required points of intersection. Then connect these points by straight 
or curved lines as required. 

Take, for instance, the vertical plane ab, in the left-hand side view. This 
will draw the trace a'a' in the top view, piercing the vertical Cylinder at a', 
which, being projected down to the front view, will intersect the traces of the 
same plane at the points a"b'', which are two points of the lines of intersection 
of the Cylinders. Any number of points can be determined in the same manner, 
and a line connecting them (in this case a straight line) will be the intersection. 
In the right-hand side view, the vertical plane cd draws the traces c'c' in the top, 
and c"c", d"d" in the front views, the points of intersection c" and d" being 
projected from c'. 

The problem consists, after all, in simply finding the projections of lines of 
different lengths, these lengths being obtained from whichever view gives ihem 
truly. 

The lines already on the drawing give all the data necessary for the 
development. 

Unroll the cylindrical surfaces into plane surfaces (as in Plate 10, Fig. 62), 
and draw upon the plane surfaces all the traces that are upon the cj'lindrical 
ones and in the same positions, atid mark the points of intersection upon these 
traces. Connect these points by lines for the development of the intersections. 



MECHANICAL DHAWlNG. 55 

Fig. 79. To draw ci similar vertical Cylinder intersected, by horizontal Cylinders of 
the dimensions and in the positions shown, and to draw the development of the entire surface. 

This is merely a modification of Fig. 78, and will best serve as an exercise 
and training if left to the ingenuity of the student without further explanation. 

PLATE 20. 

Fig. 80. To draw a vertical Cone having a base 3J" diameter and sides at 60° 
with the base, intersected by a Cylinder 1|" diameter whose axis is perpendicular to the 
side of the Cone and intersects the axis of the latter f inch above the base, the end of the 
Cylinder being 2^ from this intersection, and the cone being truncated by a plane parallel- 
to, and 2|- inches from, its base ; and to draio the development of the entire surface of 
the solid. 

Draw the top, front, and side views of the Cone and Cylinder, with the 
exception of their intersection, which is to be determined. Draw a complete 
auxiliary view of them on a plane perpendicular to the axis of the Cylinder. It 
now remains to determine their intersection. To do this, some method must be 
found in which the cutting of both Cylinder and Cone by the same plane will 
draw straight lines on the surface of each, the intersection of which lines will 
give a point of the intersection of the surfaces. As has already been explained, 
any plane which intersects a cone and at the same time passes through its apex 



56 INTERMEDIATE COURSE. 

will draw straight lines upon its surface, and any plane which intersects a 
cylinder and is parallel to its axis will draw straight lines upon its surface. 
Hence, in the present instance, if we cut the Cylinder and Cone by a series of 
planes under the above conditions, the intersections of the traces of these planes 
on the surface of the Cylinder with those on the surface of the Cone will be 
points of intersection of these surfaces. 

The Auxiliary View, being a projection upon a plane perpendicular to the 
axis of the Cylinder, presents a means of drawing the required cutting planes, 
because all planes perpendicular to it will be parallel to the axis of the Cylinder, 
and the traces of any number of such planes can be made to cut the apex a of 
the Cone. This view of the end of the Cylinder, having already been divided 
into equal parts to locate the traces upon the other views of the Cylinder which 
were used in obtaining the ellipses of its end, a convenient method will be to 
use these divisions as points through which to pass the new cutting planes. 
Hence, in this Auxiliary View, draw the trace of a plane from the apex a through 
each of these divisions. All of these planes will intersect in a line perpendicular 
to this plane of projection. Draw this line, ab, in the front view to intersect the 
plane of the base of the cone at b and project b to the top view. 

Draw a line cd, tangent to the base of the cone, in the top and auxiliary 
views, transfer the points where the cutting planes intersect this tangent in the 
auxiliary view to the tangent in the top view, and connect these points with b, 



MECHANICAL DRAWING. 57 

in the top view. Then be, bd, etc., in the top view, will be the traces of the 
cutting planes upon the horizontal plane of the base of the cone, and lines 
drawn from the" apex a to the points where these traces intersect the circle of the 
base will be the traces of the cutting planes upon the surface of the Cone. 
Project these traces to the front view, and then their intersections with the 
traces of the cutting planes already drawn upon the surface of the cylinder in 
both top and front views will be points of the line of intersection of the surfaces 
of the cone and cylinder in these views. Project these points from the front 
view to the corresponding traces on the surface of the Cylinder in the side view 
for the corresponding points in that view. 

This principle is applicable, no matter what may be the proportions or rela- 
tions of the surfaces, and, in general, if the requirement is to find the inter- 
section of any two solids, such solids must be cut by planes whose traces upon 
the surfaces of the solids will be lines, the projections of Avhich can be readily 
drawn in the different views. The fact should be remembered that the inter- 
section of the surfaces of any two solids is a line, and that, if these solids be cut 
by planes, such planes will draw lines on the surfaces of the solids which will meet 
at their intersection, if at all. The selection of the best location and arrange- 
ment of the planes is what most requires the exercise of the reasoning powers. 

To find the intersection of two cones, for instance, cutting planes can 
be used which pass through the apices of both cones, in which case the planes 



58 INTERMEDIATE COURSE. 

will draw straight lines on the surfaces of the cones, and the points where the 
lines meet will be points of the intersection. If the axes of the cones are 
parallel, cutting planes can he used which are perpendicular to these axes, in 
which case the traces upon the surfaces will be circles, and the intersections of 
the circles will be points of the intersection of the surfaces, and the projections 
of circles are almost as readily handled as those of straight lines. 

In the cases of Spheres, Ellipsoids, and surfaces of revolution, cutting planes 
perpendicular to the axis will draw circles on the surface, and no great difficulty 
need be apprehended in treating them if the principles already explained are 
understood. 

OBLIQUE AXIS OF SYMMETRY. 

Fig. 81. To draw the projections of a solid whose axis of symmetry is inclined to 
both planes of projection. 

Let the solid be a rectangular block 3 inches long, 1 inch wide, and J inch 
thick, and let the front projection of its axis of symmetry be inclined upwards at 
an angle of 60° with the horizontal plane, and the top projection inclines forwards 
at an angle of 45° with the front vertical plane. Draw the trace of a horizontal 
plane ah, and intersect it by the traces of two vertical planes cd and ef perpen- 
dicular to each other, and make the intersection of these traces, h, the lower end 



MECHANICAL DRAWING. 59 

of the inclined axis of symmetry. Draw the projections of this axis of sufficient 
length to exceed that of the block, say to g. JSTow, in order to draw the block 
of the given dimensions, it is only necessary to draw its projection on a plane 
parallel to the axis, and as h'g' is a top view of this axis, a line a'b' parallel with 
h'g' will be the trace of the horizontal plane ab upon a plane parallel with the 
axis. Hence, if g' be projected to g'" at the same vertical distance from a'b' as 
that of g from ab, then g"'h"' will be a parallel view of the axis upon which the 
side of the block can be drawn, and from which an end view can be projected 
to determine the thickness. N'ow, draw the trace of a horizontal plane Im 
through the centre of the upper end of the block, then I'm' , at the same height 
above ab, will be the trace of this same plane in the front and side vicAvs. The 
thickness being laid off on the top view, the ends can be projected from the 
auxiliary view and the top view thus completed, from which the corners can be 
projected to the front view, their vertical distances from ab and I'm' being the 
same as from a'b' and Im. The corners are then projected from the front to the 
side view, their horizontal distances from ef and rs being the same as from 
e'f and r's'. 

These traces of vertical and horizontal planes, ab, cd, ef, etc., are called, for 
brevity, centre lines, and are of great utility. They form the bases from which 
every point of an object can be definitely located, and are valuable lines of 
reference. They should always be inked blue, and preserved. 



60 INTERMEDIATE COURSE. 

Fig. 82. To draw the same rectangular block as in Fig. 81, with the front 
projection of its axis of symmetry inclining upioards at an angle of 30° with the 
horizontal plane and the top projection inclining backwards at an angle of 30° with the 
front vertical plane. 

Draw the three views of the axis, and from the top view project the axis 
upon a vertical plane parallel to it, as in Fig. 81, and upon this plane, and 
another perpendicular to it, draw the Mock in its true size, together with the 
horizontal and vertical planes of reference or centre lines. Locate these centre 
lines in the three original views, and complete the projections required. 

The change in the inclination of the block in this Figure from that in 
Fig. 81 gives a good idea of the infinite variety of positions in which an object 
can be drawn, and of the judgment required in the treatment of oblique views. 
It is evident that, no matter how any line may be inclined, a vertical plane will 
always contain it, and it can be projected upon a plane of projection parallel to 
this vertical plane. Having thus the projection of the line in its true length 
upon a vertical plane, this can be considered the front view of a new series of 
views upon which the object, however complicated it may be, can be completely 
and readily drawn, and from these views those required can be projected. 

As this is a problem of frequent occurrence in practical work, and one 
which makes a good test of a draughtsman's capacity, the student should draw 
different objects with the axes of symmetry leaning in different directions, and 
should not abandon the subject until attaining thorough familiarity with it. 



MECHANICAL DRAWING. 61 



REVIEW. 



The surface of every structure is composed of plane surfaces or curved 
surfaces, or tlie two combined. The intersections of these surfaces are either 
straight or curved lines, and the intersections of lines are points; hence, points, 
lines, and surfaces in various relations and combinations constitute everything 
that Mechanical Drawing has to deal with. 

A point has neither length, breadth, nor thickness; a line has length only; 
and a surface has length and breadth, but no thickness. A line may be con- 
sidered as made up of points, as elements, arranged according to some law; and 
a surface may be considered as made up of lines, as elements, each bearing a 
certain relation to the one adjacent to it. 

In analyzing any structure, planes of reference can be established in relation 
to it in any positions and in any number that may be desirable, and the traces 
of these plones can be drawn on several planes of projections, and the points 
and lines of the structure can be located on these planes of projection at the 
proper distances from the traces of the planes of reference, and any number of 
views of the structure thus be made. If the surface of the structure does not 
contain enough actual lines to enable it to be definitely determined, any of its 



62 INTERMEDIATE COURSE. 

elements can be assumed to be lines and treated as such. If the lines are curved, 
the points where they pierce planes of reference can be determined and various 
projections of them be constructed. 

Most structures are composed of modifications and combinations of the 
simple solids which have been investigated in this Course and the Junior Course, 
and the experience and training to be obtained from a full comprehension 
of them should serve as a complete preparation for all the problems in 
OrthdiTf^Dt^i© pi'*^jection likely to occur in practical work as applied to Engineering, 
Architecture, or any constructive art, with the exception of the Helix, which 
will be investigated in the Senior Course. 



Williams & Brown, 

N. E. Cor. Tknxh and Chestnut Sts., Philadelphia., 



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Thorne-s Mechanical Drawing Studies. 

Marks' Paper Odontograph. ^ 

Textile Design Papers. 



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